On some criteria for completely regular growth of entire functions of exponential type

2006 ◽  
Vol 80 (1-2) ◽  
pp. 114-126 ◽  
Author(s):  
V. B. Sherstyukov
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Regular Growth ◽  
Completely Regular ◽  
Growth Of Entire Functions ◽  
Functions Of Exponential Type

Growth of entire functions of exponential type defined by convolution

2006 ◽  
Vol 87 (4) ◽  
pp. 330-342 ◽  
Author(s):  
J. Müller ◽  
S. Naik ◽  
S. Ponnusamy
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Growth Of Entire Functions ◽  
Functions Of Exponential Type

scholarly journals Sufficient conditions for the improved regular growth of entire functions in terms of their averaging

2020 ◽  
Vol 12 (1) ◽  
pp. 46-54
Author(s):  
R.V. Khats'
Keyword(s):  
Entire Function ◽  
Fourier Coefficients ◽  
Entire Functions ◽  
Asymptotic Relation ◽  
Sufficient Conditions ◽  
Regular Growth ◽  
Converse Statement ◽  
Completely Regular ◽  
Growth Of Entire Functions ◽  
Finite Sum

Let $f$ be an entire function of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1<\psi_2<\ldots<\psi_m<2\pi$ and $h(\varphi)$ be its indicator. In 2011, the author of the article has been proved that if $f$ is of improved regular growth (an entire function $f$ is called a function of improved regular growth if for some $\rho\in (0,+\infty)$ and $\rho_1\in (0,\rho)$, and a $2\pi$-periodic $\rho$-trigonometrically convex function $h(\varphi)\not\equiv -\infty$ there exists a set $U\subset\mathbb C$ contained in the union of disks with finite sum of radii and such that $\log |{f(z)}|=|z|^\rho h(\varphi)+o(|z|^{\rho_1})$, $U\not\ni z=re^{i\varphi}\to\infty$), then for some $\rho_3\in (0,\rho)$ the relation \begin{equation*} \int_1^r {\frac{\log |{f(te^{i\varphi})}|}{t}}\, dt=\frac{r^\rho}{\rho}h(\varphi)+o(r^{\rho_3}),\quad r\to +\infty, \end{equation*} holds uniformly in $\varphi\in [0,2\pi]$. In the present paper, using the Fourier coefficients method, we establish the converse statement, that is, if for some $\rho_3\in (0,\rho)$ the last asymptotic relation holds uniformly in $\varphi\in [0,2\pi]$, then $f$ is a function of improved regular growth. It complements similar results on functions of completely regular growth due to B. Levin, A. Grishin, A. Kondratyuk, Ya. Vasyl'kiv and Yu. Lapenko.

Representation Formulas for Integrable and Entire Functions of Exponential Type I

1988 ◽  
Vol 40 (04) ◽  
pp. 1010-1024 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Conjugate Function ◽  
Interpolation Formula ◽  
Type I ◽  
Additional Hypothesis ◽  
Period 2 ◽  
Representation Formulas ◽  
Functions Of Exponential Type ◽  
Image Position

Let Bτ denote the class of entire functions of exponential type τ (&gt;0) bounded on the real axis. For the function f ∊ Bτ we have the interpolation formula [1, p. 143] 1.1 where t, γ are real numbers and is the so called conjugate function of f. Let us put 1.2 The function Gγ,f is a periodic function of α, with period 2. For t = 0 (the general case is obtained by translation) the righthand member of (1) is 2τGγ,f (1). In the following paper we suppose that f satisfies an additional hypothesis of the form f(x) = O(|x|-ε), for some ε &gt; 0, as x → ±∞ and we give an integral representation of Gγ,f(α) which is valid for 0 ≦ α ≦ 2.

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